Exploring the Continuity of Mathematical Functions: A Deep Dive into the Function (f(x) frac{1}{x^2 1})

Understanding the continuity of mathematical functions is a fundamental aspect of calculus and real analysis. Among these, a specific function, (f(x) frac{1}{x^2 1}), stands out as an excellent example of a continuous function. This article aims to provide a comprehensive analysis of this function, including proving its continuity using both direct observation and the concept of limits. By examining this function, readers will gain insights into the broader principles of function behavior.

What is a Continuous Function?

A function is considered continuous at a point if, for any small change in the input, the output changes by a small amount as well. Mathematically, a function (f(x)) is continuous at a point (c) if the following limit exists and equals (f(c)):

[lim_{x to c} f(x) f(c)]

This definition generalizes to the entire domain of the function, and it plays a crucial role in understanding the behavior of functions across their domains.

The Function (f(x) frac{1}{x^2 1})

The function (f(x) frac{1}{x^2 1}) is a simple yet powerful example to illustrate the concept of continuity. This function is well-defined for all real numbers (x), meaning it does not have any points where it becomes undefined.

Direct Observation

First, let's observe the function directly. Notice that the denominator (x^2 1) is always greater than zero for all real numbers (x), as (x^2 geq 0), and adding one ensures the denominator is always positive. This implies that the function (f(x)) is defined for all real numbers (x).

Using Limits to Prove Continuity

To formally prove that (f(x) frac{1}{x^2 1}) is continuous, we will use the definition of limits and the properties of continuous functions. We need to show that for any (c) in the domain of (f(x)), the limit of (f(x)) as (x) approaches (c) equals (f(c)).

Step 1: Evaluate the Function at the Point (c)

First, let's evaluate (f(x)) at a point (c):

[f(c) frac{1}{c^2 1}]

Step 2: Compute the Limit

Next, we compute the limit of (f(x)) as (x) approaches (c):

[lim_{x to c} f(x) lim_{x to c} frac{1}{x^2 1}]

To find this limit, we can directly substitute (c) into the expression:

[lim_{x to c} frac{1}{x^2 1} frac{1}{c^2 1}]

This shows that the limit of (f(x)) as (x) approaches (c) is indeed (f(c)).

Step 3: Verify Continuity

Since (lim_{x to c} f(x) f(c)), we have proven that the function (f(x) frac{1}{x^2 1}) is continuous at every point (c) in its domain.

Conclusion: Continuity of (f(x) frac{1}{x^2 1})

By demonstrating through direct observation and the use of limits, we have established that the function (f(x) frac{1}{x^2 1}) is continuous everywhere. This ensures that the function does not exhibit any abrupt changes or discontinuities throughout its domain, making it a smooth and well-behaved function.

Understanding and analyzing such functions helps in various fields, including physics, engineering, and economics, where continuous functions often model real-world phenomena. The function (f(x) frac{1}{x^2 1}) is a prime example of how simple mathematical constructs can have profound implications.